Question

help :c
possible rational zeros
f(x)=2x^3-6x^2-7x+14
please show steps

Answer 1

Hello, Jon Le!
While I can decipher your f(x)=2x^3-6x^2-7x+14, I'd encourage you to type it out using Equation Editor for greater clarity.

The most basic questions here that we need to address are what the zeros of this function are and how we'd go about finding them.

Have you been given functions like f(x)=2x^3-6x^2-7x+14 before and asked to factor them, or to determine their roots / zeros ??

Answer 2

determining the zeros

Answer 3

I can certainly help. But first I'd like to ask you what methods you've learned in the past that help you find zeros/roots/solutions where polynomials or polynomial equations are concerned.

Answer 4

i have been factoring last number and using the first term as a denominatior

Answer 5

i just started learning these so im not entirely sure what to do after

Answer 6

You're on the right track! However, it's not always the first term. It can be a factor of it.

Possible rational zeros are defined by the Rational Root Theorem.
\[\pm\frac{p}{q}\]
The set p consists of the factors of the constant of the polynomial. The set q consists of the factors of the leading coefficient of the polynomial.

For example, if you had 2x^2+3, p={3, 1} and q={2, 1}.
p/q=3, 1, 2/3, 1/3. Any of these can be either positive or negative.

Answer 7

For your case, 14 is the constant, and 2 is the leading coefficient.

Factors of 14 are 1, 2, 7, and 14.
Factors of 2 are 1 and 2.

p/q= 1/1, 2/1, 7/1, 14/1, 1/2, 2/2, 7/2, 14/2.
Simplify these and ignore duplicates.
+/- {1, 2, 7, 14, 1/2, 7/2} is your answer.

Answer 8

I like this approach. The Rational Root Theorem that tomhue is discussing is one of the methods I was asking you about for finding the roots / zeroes /solutions of polynomial equations.

Are you familiar with synthetic division?

Answer 9

i dont believe i am

Answer 10

If so, you could use synthetic div. to check each of the possible roots that tomhue has identified to see whether it actually is a root or not.

Answer 11

If you don't know synth. div., the process becomes harder. You may have to use long division; are you familiar with that?

Answer 12

yes

Answer 13

i might go on youtube and try to learn synthetic division

Answer 14

Word to the wise: synth. div. is much faster, but long div. will work.

Choose any of the possible roots / zeros that tomhue has identified.
Write the related factor using the opposite sign:

for example, if you chose possible root 2, then the related factor would be x-2. You'd then need to divide x-2 into the polynomial given, using long div. Does that seem at least familiar, if not easy?

Answer 15

yes i seen my professor do it and it looks pretty simple
ill give it a try :)

Answer 16

If you go on YouTube or anywhere else to learn synth div., I'd be very happy to help you with this when you're ready. I really like and admire your + attitude.

Answer 17

cool i might take you up on that ;)
thank you guys you helped me out a bunch <3

Answer 18

Have a minute or two more?

Answer 19

yea i do

Answer 20

Let's borrow one of the possible roots that tomhue has identified: 7. that was an arbitrary choice.

Let's set up synth. div. (which I will explain further after you've looked it up):

2x^3-6x^2-7x+14
7 | 2 -6 -7 14
14 56
---------------
2 8 49 343

Answer 21

Possibly some of this looks familiar from what your prof did in class.

Please note that the very last number, 343, is not zero; this tells us immediately that 7 is NOT a root / zero of the given polynomial.

Any questions? You'll need to look up synth div and perhaps review your class notes, and I'm sure your doing that will generate more questions, which I'd be happy to answer now or later.

Answer 22

how do we get to the number 343?

Answer 23

7 | 2 -6 -7 14
14 56
---------------
2 8 49 343

7 is the divisor (the possible root)

2, -6, -7 and 14 are the coefficients of the polynomial you were given (check this out!!

To perform the synth div, here's what I did:

Carry the 2 down below that ---- line.
Multiply that 2 by the divisor (7).
Write the result (14) under the -6.
combine -6 and 14.
Write the result below the --- line: 8.

Mult that 8 by the divisor, 7.

Write the result (56) under the -7.

Answer 24

That, in a nutshell, is a quick demo of what you do in performing synth. div.
Please see wehther you yourself can obtain that 343, following the pattern I've described above.

Answer 25

so 56-7=49
49*7=343
343+14=357

Answer 26

wait do i stop at 343 or keep going?

Answer 27

Ooops! Hats off to you for finding my mistake. Right you are.
You keep going until you have one last sum or difference, which in this particular example is 357 (as YOU calculated correctly).

Answer 28

Since this last sum or difference is NOT zero, you conclude that 7 is NOT a root or zero of the given function.

Answer 29

Wow thanks :D
i think i got this

Answer 30

this process becomes MUCH easier with practice, so I'd bet you could go thru all of tomhue's possible roots in a matter of minutes.
So nice working with you. Thanks for your persistence and hard work (and for finding my mistakes)! Hope to see you again!

Answer 31

Mind giving tomhue a medal for his important contribution to this discussion?

thanks!

And thanks for becoming my fan!

Answer 32

will do :)